Find one value of $x$ that is a solution to the equation: $(5x+2)^2+15x+6=0$ $x=$
Answer: We could solve for $x$ by expanding $(5x+2)^2$, combining terms that are alike, and using the quadratic formula or factoring to solve for $x$. However there is a shorter and more elegant way to approach this problem. Let's use structural features to rewrite the equation in a simpler form. Note that $15x+6=3({5x+2})$. This means that we can rewrite the equation as: $({5x+2})^2+3({5x+2})=0$ If we let ${p}={5x+2}$, we can see that this equation is in the form: ${p}^2+3{p}=0$ Let's solve this equation in terms of ${p}$ : $\begin{aligned}{p}^2+3{p}&=0\\\\ {p}({p}+3)&=0\\\\ {p}=0\ &\text{or} \ \ {p}=-3 \end{aligned}$ Since ${p}={5x+2}$, let's substitute this value back into our two solutions in order to solve for $x$ : ${5x+2}=0\ \ \ \text{or} \ \ \ {5x+2}=-3$ When we solve $5x+2=0$, we find that $x=-\dfrac{2}{5}$. When we solve $5x+2=-3$, we find that $x=-1$. In conclusion, the two solutions of the equation $(5x+2)^2+15x+6=0$ are $x=-\dfrac{2}{5}$ and $x=-1$. [Is there another way to solve for x?]